Understanding Differentiability of a Function

Differentiability is a fundamental concept in calculus that deals with the existence of derivatives for functions. A function is said to be differentiable at a point if it has a derivative at that point. The derivative of a function at a point measures the rate at which the function value changes as its input changes.

Checking Differentiability

To check whether a function is differentiable at a point, we need to consider the following:

1. Continuity: A function must be continuous at a point to be differentiable there. However, continuity alone does not guarantee differentiability.
2. Existence of Derivative: A function is differentiable at a point if the derivative exists at that point.

Formal Definition

A function $f(x)$ is differentiable at a point $x = a$ if the following limit exists:

$$ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} $$

If this limit exists and is finite, then $f(x)$ is differentiable at $x = a$, and $f'(a)$ is the derivative of $f(x)$ at $x = a$.

Differentiability on an Interval

A function is said to be differentiable on an open interval $(a, b)$ if it is differentiable at every point in that interval. For closed intervals $[a, b]$, we also check differentiability from the right at $a$ and from the left at $b$.

Checking Differentiability at a Point

To check the differentiability of a function at a point, follow these steps:

1. Check Continuity: Verify that the function is continuous at the point.
2. Compute Derivative: Find the derivative of the function using the definition or rules of differentiation.
3. Check Left-hand and Right-hand Derivatives: Calculate the left-hand derivative (LHD) and right-hand derivative (RHD) at the point.
4. Compare LHD and RHD: If LHD and RHD are equal, the function is differentiable at that point.

Left-hand and Right-hand Derivatives

The left-hand derivative (LHD) at $x = a$ is defined as:

$$ LHD = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h} $$

The right-hand derivative (RHD) at $x = a$ is defined as:

$$ RHD = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h} $$

If both LHD and RHD exist and are equal, then the function is differentiable at $x = a$.

Table of Differences and Important Points

Aspect	Continuity	Differentiability
Definition	A function is continuous at a point if the limit of the function as it approaches the point is equal to the function's value at that point.	A function is differentiable at a point if it has a defined derivative at that point.
Requirement	A function must be continuous at a point to be differentiable there.	Differentiability implies continuity, but not vice versa.
Graphical Interpretation	No sudden jumps or breaks in the graph.	The graph has a tangent at the point, and it is not a sharp corner or cusp.

Examples

Example 1: Differentiable Function

Consider the function $f(x) = x^2$. To check its differentiability at $x = a$, we compute the derivative:

$$ f'(x) = \frac{d}{dx}(x^2) = 2x $$

Since the derivative exists for all $x$ in the domain of $f(x)$, the function is differentiable everywhere.

Example 2: Non-differentiable Function

Consider the function $f(x) = |x|$. To check its differentiability at $x = 0$, we compute the LHD and RHD:

$$ LHD = \lim_{h \to 0^-} \frac{|0 + h| - |0|}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1 $$

$$ RHD = \lim_{h \to 0^+} \frac{|0 + h| - |0|}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1 $$

Since LHD $\neq$ RHD, the function $f(x) = |x|$ is not differentiable at $x = 0$.

By understanding these concepts and applying the steps outlined above, one can determine the differentiability of functions, which is crucial for further analysis in calculus, such as optimization and curve sketching.